Question

Find the odd one among : $6,19,56,169,508,1519$.
(A)$19$
(B)$169$
(C)$508$
(D)$1519$

Answer 1

Hint: For such problems the series must satisfy some pattern or the series is based on some mathematical sequence. Find the pattern and the odd one will be that one which fails to satisfy the pattern. To start solving, we can consider the first two terms, 6 and 19. We can observe that 19 is one more than the third multiple of 6.

Complete step-by-step solution -
We have to find the odd one among : $6,19,56,169,508,1519$.
That is we have to find the pattern first which satisfies this series.
We can see that $6\times 3+1=18+1=19$.
So if we multiply the first term $6$ by $3$ and then add $1$ with the result we get the second term $19$.
Now $19\times 3-1=57-1=56$.
So if we multiply the second term $19$ by $3$ and subtract $1$ from the result we get the third term $56$.
Also $56\times 3+1=168+1=169$.
So we can see that if we multiply the third term $56$ by $3$ and then add $1$ with the result we get the fourth term $169$.
In the similar way $169\times 3-1=507-1=506$.
So to satisfy this pattern the fifth term must be $506$.
But in the given series the fifth term is $508$.
Now take the fifth term $506$ and using the same pattern.
Then the sixth term $=506\times 3+1=1518+1=1519$.
From the above pattern that we have got, we get the terms as
First term $\times 3+1=$ second term
Second term $\times 3-1=$ third term
Third term $\times 3+1=$ fourth term
Fourth term $\times 3-1=$ fifth term
Fifth term $\times 3-1=$ sixth term
Therefore the given series will satisfy the same pattern if the fifth term is $506$.
But as in the given series fifth term is $508$ so $508$ is the odd one in the series.
Therefore the odd one among : $6,19,56,169,508,1519$ is $508$.
So the correct answer is option (c).

Note: In this type of questions, the key is to identify the pattern in which the terms are arranged. The sooner we are able to decode it, the sooner we can answer the question. So, we must always start by comparing the first 2 terms, trying operations like addition, subtraction, multiplication and division. We can check if they are coming under a particular category like even, odd, prime, multiples of a numbers, factors of a number and so on. Here, we might get confused when comparing second and third terms as it does not follow the same pattern, we must not stop there. We must observe for a small change also, just like we had +1 and -1 in this case.